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Physica Scripta

Considerations Concerning the Harmonic Limit of the Morse Oscillator

Dušan Popov

Published Apr 1, 2001• 25 citations
Publisher

Abstract

The comprehensive conditions concerning the harmonic limit of the Morse oscillator are investigated in this paper. These conditions extend the over-simple “classical” condition, which results by truncating the expansion of the Morse potential function in the power series of the coordinate close to the equilibrium position. The validity of the obtained results is tested by calculating the harmonic limit of the expected values of some characteristic physical observables of the Morse oscillator systems.

DOI
10.1238/physica.regular.063a00257

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The comprehensive conditions concerning the harmonic limit of the Morse oscillator are investigated in this paper. These conditions extend the over-simple “classical” condition, which results by truncating the expansion of the Morse potential function in the power series of the coordinate close to the equilibrium position. The validity of the obtained results is tested by calculating the harmonic limit of the expected values of some characteristic physical observables of the Morse oscillator systems.
We obtain the exact, normalized, closed-form eigenfunctions for the one-dimensional Morse oscillator, as well as the raising and lowering operators. We next review a new method for obtaining the coherent states for arbitrary potentials, it being based on the classical motion of a particle. We apply this method to the Morse oscillator. After demonstrating that in the appropriate limit our results reduce to those for the harmonic oscillator, we obtain analytic insights into published numerical results on uncertainty products for Morse-oscillator wave packets.
Quantum calculations of the dynamics of a Morse oscillator with a Gaussian initial wave packet have been performed. As previously observed in other simple oscillators the classical and quantum expectation values for the coordinate exhibit significantly different behaviors, especially after the first few cycles. Some theories of chemical kinetics require the classical-mechanical calculation of lifetimes which may be quite long compared to vibrational periods. These theories deserve re-examination in view of the dynamical behavior observed in the present work.
A realization of the raising and lowering operators for the Morse potential is presented. We show that these operators satisfy the commutation relations for the SU(2) group. Closed analytical expressions are derived for the matrix elements of different functions such as 1/y and d/dy. The harmonic limit of the SU(2) operators is also studied. The transition probability between two eigenstates produced by a harmonic perturbation as a function of the operators [Formula: see text]±,0 is discussed. The average values of some observables in the coherent states |α > for the Morse potential are also calculated. PACS Nos.: 02.30+b, 03.65Fd, 42.50Ar, and 33.10Cs
Quantum relativistic solutions for a particle in a one-dimensional Morse potential are presented using methods previously proposed and applied to a particle in a harmonic oscillator. The methods lead to both numerical and analytical solutions, with the latter allowing smooth variation of the system parameters from non-relativistic to ultrarelativistic limits. Analytical expressions for the energy levels and wavefunctions are obtained, as solutions to a Schrödinger-type equation, including relativistic effects through a state dependent rescaled mass. The eigenstates of the Morse potential exhibit suitable and smooth behavior, and approach the corresponding harmonic oscillator solutions as the depth of the Morse potential well increases, as expected. A comparison is also presented between the relativistic harmonic oscillator obtained with this method, and the so-called Klein-Gordon oscillator.
The Morse potential quantum system is a realistic model for studying vibrations of atoms in a diatomic molecule. This system is very close to the harmonic oscillator one. We thus propose a construction of squeezed coherent states similar to the one of harmonic oscillator using ladder operators. Properties of these states are analysed with respect to the localization in position, minimal Heisenberg uncertainty relation, the statistical properties and illustrated with examples using the finite number of states in a well-known diatomic molecule.